- Topic
- Taylor polynomials
- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 4
Taylor polynomials practice question 13
Find the third-order Taylor polynomial of about , where is the natural logarithm.
is a product of two functions of . Which rule does the first derivative need, and is there still a product after that?
Product rule first: . The next two derivatives are then quick.
The term is , and .
A polynomial in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
Differentiate three times, using the product rule once at the start, then evaluate at the centre 1, where log 1 = 0, and divide each derivative by its factorial.
Working
Product rule: both x and log x vary.
After the first derivative the product is gone.
Evaluate at the centre; log 1 = 0.
Divide by 2! and 3!. As f(1) = 0, there is no constant term.
Answer
The answer is T three of x equals (x minus 1), plus one half of (x minus 1) squared, minus one sixth of (x minus 1) cubed.
Check
At x = 1.1, T₃ gives 0.10483 and 1.1 log 1.1 ≈ 0.10484.
Where this goes wrong
Differentiating x log x as x · (1/x) = 1 and ending with T₃(x) = x − 1. Both factors vary, so the product rule gives log x + 1, whose own derivatives 1/x and −1/x² are not 0.
Leans on
The product rule, from Differentiation rules
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